37 courses in the 1981–82 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
Applied Mathematics Course Descriptions
Students)
Review of basic algebra, geometry, and trigonometry. Sets, relations, operations, and
_ functions. Complex numbers. Exponential and
logarithmic functions. Difference equations. Interpolation, estimation, and approximation. Introduction to linear algebra and matrix theory. Prereq: None : N.B.: Not open to students in the Faculty of Mathematics
Students) . An introduction to probability and statistics. Deterministic and non-deterministic modelling. Numerical methods using a pocket calculator. Prereq: AM 101, or consent of instructor.
N.B.: Not open to students in the Faculty of Mathematics.
AM 230, 240, 260, 270, 340, 430, 440 is a sequence of courses designed to illustrate the representation of a real situation by a mathematical model. The mathematical techniques necessary to formulate the model and to discuss its behaviour are developed as required. Examples will be taken from diverse fields since one of the aims of each course is to formulate models to describe unfamiliar situations.
Ordinary differential equations will be applied to models drawn from biology, economics, physiology and physics.
Prereq: Math 120a/b
N.B.: Credit will be given for only one of AM 230, 260.
Other examples from biology, ecology, economics, physiology and physics will be discussed using sets of differential equations.
Prereq: AM 230 or Math 220b
N.B.: Credit will be given for only one of AM 240.
Mathematical models for problems in the physical and biological sciences. Typical problems chosen from ecology, special relativity, spread of epidemics, rumours and tumors. Solutions to problems will be obtained primarily by differential equations.
Prereq: Math 130a/b.
N.B. Credit will be given for only one of AM 230, 260.
Further mathematical models from various disciplines. An introduction to Newtonian mechanics will also be included in the course.
Prereq: AM 260 or consent of instructor.
N.B.: Credit will be given for only one of AM 240, 270.
Difference equations, Laplace transforms applied t. to discrete and continuous mathematical models taken from ecology, biology, economics and other fields. Prereq: Math 220a/b or consent of instructor.
Curves in Euclidean 3-space (E*) and the Serret- Frenet formulae; surfaces in E? and their intrinsic geometry. Gaussian curvature and the Gauss-Bonnet theorem. Co-ordinate transformations and tensors in ‘ n-dimensions; n-dimensional Riemannian spaces; covariant differentiation; geodesics; the curvature, Ricci and Einstein tensors. Applications of tensors in Relativity and Continuum Mechanics.
Prereq: Math 230a/b, or consent of instructor.
N.B.: Cross listed with PMath 365.
A tensorial formulation of special relativity, Lorentz transformations and flat space-time; relativistic mechanics and collision phenomena; Maxweil’s equations in tensorial form and some of their solutions; energy-momentum tensor of the electromagnetic field and of a fluid; spinors and representations of the Lorentz group. Reference will be made throughout to the experimental! verification of special relativity.
Prereq: AM 362/P Math 365, Phys 364, or consent of instructor.
- Course Descriptions
Applied Mathematics
Stress and Strain tensors: analysis of stress and Strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and energy, constitutive equations. Navier-Stokes equation. Basic equations of elasticity. Various applications.
Prereq: Math 230a/b, or consent of instructor.
First order partial differential equations and methods of characteristics, second orderpartial differential equations, boundary value problems and related numerical methods, vibrating string, 2-D membranes, heat equation and related problems; introduction to vector analysis.
Prereq: AM 260, Math 230a/b, or consent of instructor..
Existence and uniqueness theorems, second and higher order equations, series solutions and Special Functions, Laplace transforms. Application to Mathematical Physics.
Prereq: Math 230a/b.
Sturm-Liouville problems and eigenfunction expansions. Non-linear mechanics, phase plane method and stability analysis.
Prereq: AM 381.
Three-dimensional rigid body motion. Lagrange’s equations of motion. Ignorable co-ordinates. Hamilton’s principle. Noether’s theorem and conservation laws. Canonical equations. Canonical transformations. Lagrange and Poisson bracket relations. Integral invariants.
Prereq: Math 230a/b, or consent of instructor.
Ordinary and partial differential equations. Laplace and Fourier transforms. Special functions. Fourier series. Vector and matrix methods. Calculus of variations. Other topics in advanced calculus.’ Prereq: Consent of instructor.
N.B.: Not open to students in the Faculty of Mathematics.
Integral equations and integral! transforms will be applied to systems with memory.
Prereq: Consent of instructor.
As a project, students will develop a mathematical model and interpret its behaviour.
Prereq: Consent of instructor.
Non-linear mechanics, stability, quasi-linear and strongly non-linear systems, linear periodic systems, non-linear integral equations.
Prereq: AM 381/391, or consent of instructor.
The theory of measure and the Lebesgue integral. Prereq: Math 332a or PMath 351a.
Differentiable manifolds, vector fields, linear connections, tensor fields, differential forms, and the Cartan structure equations.
Prereq: AM 362 or consent of instructor.
N.B.: Cross listed with PMath 463. -
Curved space-time and the Einstein field equations; the Schwarzschild solution; experimental tests of general relativity; the weak field limit; introduction to black holes; construction of stellar models; introduction to cosmology; 3+ 1 decomposition of space-time and the Cauchy initial value problem; methods of solving the field équations.
Prereq: AM 364, or AM 362 and consent of instructor.
Course Descriptions 381 Applied Mathematics
Schrédinger equation. Solution of one dimensional problems, commutation relations, uncertainty principle, W.K.B. approximation. Hydrogen atom, angular momentum. Scattering theory. Matrix mechanics, symmetry groups.
Prereq: Math 231a/b, AM 371 or consent of instructor.
Fundamental equations of inviscid fluids, compressibility, vorticity; two and three-dimensional irrotational, incompressible flow, Blasius’ theorem, Joukowski hypothesis. Water Wave Motion.
Prereq: AM 365.
A selection of special! topics given by members of the Applied Mathematics Department.
Prereq: Consent of instructor.
Linear operators in Hilbert spaces. Compact operators, Introduction to functional analysis. Prereq: AM 462 or consent of instructor.
Depending on instructor, topics from pseudo- Riemannian and Riemannian geometry such as: isometries and Killing vector fields; sectional curvatures; submanifolds; normal co-ordinates; variation of geodesics and Jacobi fields; conformally related spaces.
Prereq: P Math 463/AM 463 or consent of instructor. N.B.: Cross listed with P Math 465
Shock wave theory, supersonic flow around a corner, Prandtl-Meyer flow. Dynamics of real fluids, Navier- Stokes equations, exact solutions, Stokes and © Oseen flow; Turbulence, Boundary layer theory. Introduction to Geophysical Fluid Dynamics.
Prereq: AM 466.
Same as in AM 468.
Prereq: Consent of instructor.
AM 481: Y 2C 1.0
Partial Differential Equations of Applied Mathematics 2
Second-order partial differential equations and . ‘characteristics; d’'Alemberts’ solution of the wave equation, concepts of distributions, construction of Green’s functions, Fourier integral theorem, integral transforms, integral equations, variational properties of eigenvalues and eigenfunctions, special functions, asymptotic series. All these methods are applied to physical problems.
Prereq: AM 371, 381, 391, or consent of instructor.
Euler-Lagrange equations for constrained and unconstrained single and double integral variational problems. Parameter-invariant single integrals. General variational formula. The canonical formalism. Hilbert’s independent integral. Hamilton- Jacobi equation and the Caratheodory complete figure. Fields and the Legendre and Weierstrass sufficient conditions.
Prereq: Math 230a/b, or consent of instructor.
Applications of Maxwell's equations. Introduction to wave guides and antennae.
Prereq: Phys 253 or consent of instructor.
Applications of probability theory to theoretical Physics.
Prereq: Consent of instructor.
Dynamical systems, controllability and observability. Minimization of functions and functionals. Optimal control. Pontryagin’s Maximum Principle.
Prereq: Consent of instructor.
Basic equations of elasticity for homogeneous isotropic bodies; bending of beams; plane elastic waves; Rayleigh surface waves, Love waves. Solution of problems by potentials, variational methods and Saint Venants’ principle.
Prereq: AM 365.
Course Descriptions Applied Mathematics